Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Lesson 8: Adding and Subtracting Polynomials
Classwork
Exercise 1
a.
How many quarters, nickels, and pennies are needed to make $1.13?
b.
Fill in the blanks:
8,943 = _______ × 1000 + _______ × 100 + _______ × 10 + _______ × 1
= _______ × 103 + _______ × 102 + _______ × 10 + _______ × 1
c.
Fill in the blanks:
8,943 = _______ × 203 + _______ × 202 + _______ × 20 + _______ × 1
d.
Fill in the blanks:
113 = _______ × 52 + _______ × 5 + _______ × 1
Exercise 2
Now let’s be as general as possible by not identifying which base we are in. Just call the base 𝑥.
Consider the expression 1 ∙ 𝑥 3 + 2 ∙ 𝑥 2 + 7 ∙ 𝑥 + 3 ∙ 1, or equivalently 𝑥 3 + 2𝑥 2 + 7𝑥 + 3.
a.
What is the value of this expression if 𝑥 = 10?
b.
What is the value of this expression if 𝑥 = 20?
Lesson 8:
Adding and Subtracting Polynomials
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Exercise 3
a.
When writing numbers in base 10, we only allow coefficients of 0 through 9. Why is that?
b.
What is the value of 22𝑥 + 3 when 𝑥 = 5? How much money is 22 nickels and 3 pennies?
c.
What number is represented by 4𝑥 2 + 17𝑥 + 2 if 𝑥 = 10?
d.
What number is represented by 4𝑥 2 + 17𝑥 + 2 if 𝑥 = −2 or if 𝑥 = ?
e.
What number is represented by −3𝑥 2 + √2𝑥 +
2
3
1
when 𝑥 = √2?
2
POLYNOMIAL EXPRESSION: A polynomial expression is either
1.
A numerical expression or a variable symbol, or
2.
The result of placing two previously generated polynomial expressions into the blanks of the addition operator
(__+__) or the multiplication operator (__×__).
Lesson 8:
Adding and Subtracting Polynomials
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
S.35
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M1
ALGEBRA I
Exercise 4
Find each sum or difference by combining the parts that are alike.
a.
417 + 231 = _____ hundreds + _____ tens + _____ ones + _____ hundreds + _____ tens + _____ ones
= _____ hundreds + _____ tens + _____ ones
b.
(4𝑥 2 + 𝑥 + 7) + (2𝑥 2 + 3𝑥 + 1)
c.
(3𝑥 3 − 𝑥 2 + 8) − (𝑥 3 + 5𝑥 2 + 4𝑥 − 7)
d.
3(𝑥 3 + 8𝑥) − 2(𝑥 3 + 12)
e.
(5 − 𝑡 − 𝑡 2 ) + (9𝑡 + 𝑡 2 )
f.
(3𝑝 + 1) + 6(𝑝 − 8) − (𝑝 + 2)
Lesson 8:
Adding and Subtracting Polynomials
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Lesson Summary
A monomial is a polynomial expression generated using only the multiplication operator (__×__). Thus, it does not
contain + or − operators. Monomials are written with numerical factors multiplied together and variable or other
symbols each occurring one time (using exponents to condense multiple instances of the same variable).
A polynomial is the sum (or difference) of monomials.
The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial.
The degree of a polynomial is the degree of the monomial term with the highest degree.
Problem Set
1.
Celina says that each of the following expressions is actually a binomial in disguise:
i.
5𝑎𝑏𝑐 − 2𝑎2 + 6𝑎𝑏𝑐
ii.
5𝑥 3 ∙ 2𝑥 2 − 10𝑥 4 + 3𝑥 5 + 3𝑥 ∙ (−2)𝑥 4
iii.
(𝑡 + 2)2 − 4𝑡
iv.
5(𝑎 − 1) − 10(𝑎 − 1) + 100(𝑎 − 1)
v.
(2𝜋𝑟 − 𝜋𝑟 2 )𝑟 − (2𝜋𝑟 − 𝜋𝑟 2 ) ∙ 2𝑟
For example, she sees that the expression in (i) is algebraically equivalent to 11𝑎𝑏𝑐 − 2𝑎2 , which is indeed a
binomial. (She is happy to write this as 11𝑎𝑏𝑐 + (−2)𝑎2 , if you prefer.)
Is she right about the remaining four expressions?
2.
3.
4.
Janie writes a polynomial expression using only one variable, 𝑥, with degree 3. Max writes a polynomial expression
using only one variable, 𝑥, with degree 7.
a.
What can you determine about the degree of the sum of Janie’s and Max’s polynomials?
b.
What can you determine about the degree of the difference of Janie’s and Max’s polynomials?
Suppose Janie writes a polynomial expression using only one variable, 𝑥, with degree of 5, and Max writes a
polynomial expression using only one variable, 𝑥, with degree of 5.
a.
What can you determine about the degree of the sum of Janie’s and Max’s polynomials?
b.
What can you determine about the degree of the difference of Janie’s and Max’s polynomials?
Find each sum or difference by combining the parts that are alike.
a.
(2𝑝 + 4) + 5(𝑝 − 1) − (𝑝 + 7)
f.
(12𝑥 + 1) + 2(𝑥 − 4) − (𝑥 − 15)
b.
(7𝑥 4
g.
(13𝑥 2 + 5𝑥) − 2(𝑥 2 + 1)
h.
(9 − 𝑡 − 𝑡 2 ) − (8𝑡 + 2𝑡 2 )
i.
(4𝑚 + 6) − 12(𝑚 − 3) + (𝑚 + 2)
j.
(15𝑥 4 + 10𝑥) − 12(𝑥 4 + 4𝑥)
4
c.
+ 9𝑥) − 2(𝑥 + 13)
(6 − 𝑡 − 𝑡 4 ) + (9𝑡 + 𝑡 4 )
d.
(5 − 𝑡 2 ) + 6(𝑡 2 − 8) − (𝑡 2 + 12)
e.
(8𝑥 3 + 5𝑥) − 3(𝑥 3 + 2)
Lesson 8:
Adding and Subtracting Polynomials
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
3
2
S.37
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.